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Magnetic susceptibility of percolating clusters
Volume 87A, number 1,2
PHYSICS LETTERS
21 December 1981
MAGNETIC SUSCEPTIBILITY OF PERCOLATING CLUSTERS ~ Michael J. STEPHEN Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA
Received 20 October 1981 The diamagnetic susceptibility of a superconducting percolating cluster is related to the fraction of superconducting links in connected loops and the effective area of these ioops in the cluster. The susceptibility is evaluated for a simple twodimensional fractal model of a cluster. The average susceptibility ispredicted not to diverge at the percolation point.
Recently the diamagnetism of percolating clusters composed of superconducting grains has been discussed by de Gennes [1]. The diamagnetic si~isceptibiityof a cluster is related to the fraction of grains in connected loops and the effective area of these loops. For normal clusters an analog to the diamagnetism exists. We consider a two-dimensional cluster composed of normal resistors and apply a magnetic field which increases linearly with time perpendicular to the cluster. This field induces a steady emf in each closed loop proportional to the area of the loop. Omitting all dangling ends, we choose a set of independent loops in the cluster which we label by the subscripts j, k. IfEk and .J~ are the emf and current in loop k the circuit equations are =
~ RjkJk,
(1)
k
whereRkk is the resistance of loop k and Ryk is the resistance common to loops! and k. The total dissipation in the cluster is DT
=
~
=
~I~R~1Ek.
We will assume that
=
=
C
1Ak, ~ AIL7~ jk
(3)
whereL~is the total length of all the ioops in the cluster (lengths in common are counted once). The dissipation per unit length is then D = (E2/p)S. In the superconducting case the diamagnetic susceptibility per unit length is x = (A~/l6ir2X2)S where X is the penetration depth andA~is the cross-sectional area of the grains. In general it is difficult to evaluate (3). However, recently a simple fractal model for the backbone of a percolating cluster has been proposed by Gefen et al. [2] . This fractal, known as Sierpinski’s gasket, is shown in fig. 1. The critical exponents calculated with this model by Gefen et al. agree well with other estimates for percolating systems of low dimensionality. We construct the fractal by doubling its length scale at each step so that the emfE induced in a basic triangle and the resistance p of a side of such a triangle are con-
(2)
EA
1 where A1 is the area of loop I and Rik = pLik where Lik is the length common to loopsj and k. Then, following de Gennes, we define an effective area per unit connected length of
~‘
1 S
Supported in part by the NSF under Grant No. DMR81 06151.
__________
______
Fig. 1. The fractal, Sierpinski’s gasket, at the n fractal dimension isD = ln 3/ln 2 1.585.
=
3 stage. The
67
Volume 87A, number 1,2
PHYSICS LETTERS
stant. At the nth stage the total area isA,
21 December 1981
7 = 4°in units of the basic triangle, the total length of the circuit is Lcn = 3n+1 and Gefen et a). have shown that the resistancebetween two vertices of the triangle is 3~ 2 R — 1 1n (4~\ ~ ~ ‘~ ‘ ‘ ‘ The fractal contains all sizes of loops up to the scale n and the nth fractal can be constructed by joining 3 fractals of order n—i at their vertices producing a simple central triangle ofareaAn_i. Ifwe superpose the solutions of (1) would for thehave 3 fractals of—EA order 1, the central triangle an emf 12 n —instead of +EA~_ 1.This is corrected by having an extra emf2EA0_1andacurrentJ=(2A~_1/3R01)Ein the central triangle. This leads to the recursion relation for the dissipation 2 IR )F2 D =3D +~(A n n—i 3 n—lI ri—i
This is related to the exponent v introduced by de Gennes [1] by vy~= (~ + ~) ~‘• In order to calculate the average dissipation per bond in a lattice we need to know the fraction of bonds in a cluster which lie in the backbone of the cluster. We will assume that this is the same for finite clusters as for the infinite cluster and thus is proportional to L (~0)/~where ~‘ is the backbone exponent x introduced by de Gennes by ~3’-~ = x (~3+ The average number 2~3+~t)/v of clusters size L all perthis bond and of putting to- is proportional to L ( gether the susceptibility per bond is
This is easily solved to give 2/p) [(i5)n-t + ~]. (6) S~=D~/L~~ =~ (E This result shows that the dissipation and susceptibility in this model are dominated by the contribution of the largest loop. We define an exponent by 5 ‘-~J~S where L ~ 2’~is the length scale. Then
ed not to diverge at the percolation point in this model. References
=
68
in ~/ln 2
1.68.
(7)
~
~o~’~— ~‘—~—~‘r)/v
~
(8
~ p1 where we have assumed 1. ~, the correlation length. In two v~ 1.35dimensions (Stauffer[3I)andusingec ~‘ 0.5, ~ 0.14, y 2.4 and 1.(7):~3’i-~+y 0.77 and the susceptibility per bond is predict-
iii PG. de Gennes, C.R. Acad. Sci. Paris 292 (1981) 11-9. 121 Y. Gefen, A. Aharony, B.B. Mandelbrot and S. Kirkpatrick, to be published. See also Y. Gefen, B.B. Mandelbrot and A. Aharony, Phys. Rev. Lett. 45 (1980) 855. 131 D. Stauffer, Phys. Rep. 54 (1979) 3.
PHYSICS LETTERS
21 December 1981
MAGNETIC SUSCEPTIBILITY OF PERCOLATING CLUSTERS ~ Michael J. STEPHEN Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA
Received 20 October 1981 The diamagnetic susceptibility of a superconducting percolating cluster is related to the fraction of superconducting links in connected loops and the effective area of these ioops in the cluster. The susceptibility is evaluated for a simple twodimensional fractal model of a cluster. The average susceptibility ispredicted not to diverge at the percolation point.
Recently the diamagnetism of percolating clusters composed of superconducting grains has been discussed by de Gennes [1]. The diamagnetic si~isceptibiityof a cluster is related to the fraction of grains in connected loops and the effective area of these loops. For normal clusters an analog to the diamagnetism exists. We consider a two-dimensional cluster composed of normal resistors and apply a magnetic field which increases linearly with time perpendicular to the cluster. This field induces a steady emf in each closed loop proportional to the area of the loop. Omitting all dangling ends, we choose a set of independent loops in the cluster which we label by the subscripts j, k. IfEk and .J~ are the emf and current in loop k the circuit equations are =
~ RjkJk,
(1)
k
whereRkk is the resistance of loop k and Ryk is the resistance common to loops! and k. The total dissipation in the cluster is DT
=
~
=
~I~R~1Ek.
We will assume that
=
=
C
1Ak, ~ AIL7~ jk
(3)
whereL~is the total length of all the ioops in the cluster (lengths in common are counted once). The dissipation per unit length is then D = (E2/p)S. In the superconducting case the diamagnetic susceptibility per unit length is x = (A~/l6ir2X2)S where X is the penetration depth andA~is the cross-sectional area of the grains. In general it is difficult to evaluate (3). However, recently a simple fractal model for the backbone of a percolating cluster has been proposed by Gefen et al. [2] . This fractal, known as Sierpinski’s gasket, is shown in fig. 1. The critical exponents calculated with this model by Gefen et al. agree well with other estimates for percolating systems of low dimensionality. We construct the fractal by doubling its length scale at each step so that the emfE induced in a basic triangle and the resistance p of a side of such a triangle are con-
(2)
EA
1 where A1 is the area of loop I and Rik = pLik where Lik is the length common to loopsj and k. Then, following de Gennes, we define an effective area per unit connected length of
~‘
1 S
Supported in part by the NSF under Grant No. DMR81 06151.
__________
______
Fig. 1. The fractal, Sierpinski’s gasket, at the n fractal dimension isD = ln 3/ln 2 1.585.
=
3 stage. The
67
Volume 87A, number 1,2
PHYSICS LETTERS
stant. At the nth stage the total area isA,
21 December 1981
7 = 4°in units of the basic triangle, the total length of the circuit is Lcn = 3n+1 and Gefen et a). have shown that the resistancebetween two vertices of the triangle is 3~ 2 R — 1 1n (4~\ ~ ~ ‘~ ‘ ‘ ‘ The fractal contains all sizes of loops up to the scale n and the nth fractal can be constructed by joining 3 fractals of order n—i at their vertices producing a simple central triangle ofareaAn_i. Ifwe superpose the solutions of (1) would for thehave 3 fractals of—EA order 1, the central triangle an emf 12 n —instead of +EA~_ 1.This is corrected by having an extra emf2EA0_1andacurrentJ=(2A~_1/3R01)Ein the central triangle. This leads to the recursion relation for the dissipation 2 IR )F2 D =3D +~(A n n—i 3 n—lI ri—i
This is related to the exponent v introduced by de Gennes [1] by vy~= (~ + ~) ~‘• In order to calculate the average dissipation per bond in a lattice we need to know the fraction of bonds in a cluster which lie in the backbone of the cluster. We will assume that this is the same for finite clusters as for the infinite cluster and thus is proportional to L (~0)/~where ~‘ is the backbone exponent x introduced by de Gennes by ~3’-~ = x (~3+ The average number 2~3+~t)/v of clusters size L all perthis bond and of putting to- is proportional to L ( gether the susceptibility per bond is
This is easily solved to give 2/p) [(i5)n-t + ~]. (6) S~=D~/L~~ =~ (E This result shows that the dissipation and susceptibility in this model are dominated by the contribution of the largest loop. We define an exponent by 5 ‘-~J~S where L ~ 2’~is the length scale. Then
ed not to diverge at the percolation point in this model. References
=
68
in ~/ln 2
1.68.
(7)
~
~o~’~— ~‘—~—~‘r)/v
~
(8
~ p1 where we have assumed 1. ~, the correlation length. In two v~ 1.35dimensions (Stauffer[3I)andusingec ~‘ 0.5, ~ 0.14, y 2.4 and 1.(7):~3’i-~+y 0.77 and the susceptibility per bond is predict-
iii PG. de Gennes, C.R. Acad. Sci. Paris 292 (1981) 11-9. 121 Y. Gefen, A. Aharony, B.B. Mandelbrot and S. Kirkpatrick, to be published. See also Y. Gefen, B.B. Mandelbrot and A. Aharony, Phys. Rev. Lett. 45 (1980) 855. 131 D. Stauffer, Phys. Rep. 54 (1979) 3.